Problem: Determine how many solutions exist for the system of equations. ${2x-y = 7}$ ${-6x+3y = -30}$
Convert both equations to slope-intercept form: ${2x-y = 7}$ $2x{-2x} - y = 7{-2x}$ $-y = 7-2x$ $y = -7+2x$ ${y = 2x-7}$ ${-6x+3y = -30}$ $-6x{+6x} + 3y = -30{+6x}$ $3y = -30+6x$ $y = -10+2x$ ${y = 2x-10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x-7}$ ${y = 2x-10}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.